Reducing orbital eccentricity in binary black hole simulations

TL;DR

This work addresses residual eccentricity in binary black hole simulations by extending the quasi-equilibrium initial-data framework to include nonzero radial velocity via the extended conformal thin sandwich (XCTS) formalism. The authors iteratively tune the orbital frequency $\Omega_0$ and radial velocity $v_r$ to minimize eccentricity, achieving low-eccentricity evolutions over about five orbits and showing that gravitational-wave content remains largely unchanged relative to quasi-circular data. After compensating for time and phase shifts, the waveforms from low-eccentricity and quasi-circular runs agree with overlaps around $0.99$, indicating that eccentricity in QC initial data is not a limiting factor for short, high-precision GW modeling in this regime. The methodology lays groundwork for extending low-eccentricity initial data to unequal-m mass and spinning binaries, improving accuracy for GW event detection and parameter estimation.

Abstract

Binary black hole simulations starting from quasi-circular (i.e., zero radial velocity) initial data have orbits with small but non-zero orbital eccentricities. In this paper the quasi-equilibrium initial-data method is extended to allow non-zero radial velocities to be specified in binary black hole initial data. New low-eccentricity initial data are obtained by adjusting the orbital frequency and radial velocities to minimize the orbital eccentricity, and the resulting ($\sim 5$ orbit) evolutions are compared with those of quasi-circular initial data. Evolutions of the quasi-circular data clearly show eccentric orbits, with eccentricity that decays over time. The precise decay rate depends on the definition of eccentricity; if defined in terms of variations in the orbital frequency, the decay rate agrees well with the prediction of Peters (1964). The gravitational waveforms, which contain $\sim 8$ cycles in the dominant l=m=2 mode, are largely unaffected by the eccentricity of the quasi-circular initial data. The overlap between the dominant mode in the quasi-circular evolution and the same mode in the low-eccentricity evolution is about 0.99.

Figures (9)

• Figure 1: Evolution of quasi-circular initial data. The left panel shows the proper separation $s$ between the apparent horizons, computed at constant coordinate time along the coordinate line connecting the centers of the horizons, and the right panel shows its time derivative $ds/dt$. This evolution was run at three different resolutions, with the medium and high resolution tracking each other very closely through the run.
• Figure 2: Evolution of the orbital phase. The main panel shows the phase of the trajectories of the centers of the apparent horizons as a function of time for the quasi-circular (dotted curves) and low-eccentricity (solid curves) initial data. The top left inset shows the phase differences between different resolution runs, which decreases at higher resolutions. The lower right inset shows the difference in the orbital phase between evolutions with different outer boundary locations.
• Figure 3: Radial velocity during evolutions of quasi-circular and low-eccentricity initial data. The left panel shows the coordinate velocity $\dot{d}(t)$, the right panel the velocity determined from the intra-horizon proper separation $\dot{s}(t)$.
• Figure 4: Trajectories of the center of the apparent horizons in asymptotically inertial coordinates for the runs E1 (left plot) and QC (right plot). The solid/dashed line distinguish the two black holes; the circles and ellipsoids in the left figure denote the location of the apparent horizon at the beginning and end of the evolution.
• Figure 5: Proper separation (left) and orbital frequency (right) for evolutions of the QC and F initial data. The lower panels show the differences between the time-shifted QC and the F2 runs. The dotted lines in the lower panels show the differences between the E1 and F2 runs, providing an estimate of the remaining eccentricity in the F2 run.